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author | Bertram Felgenhauer <int-e@gmx.de> | 2005-08-22 16:29:56 +0000 |
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committer | Bertram Felgenhauer <int-e@gmx.de> | 2005-08-22 16:29:56 +0000 |
commit | c7a35fbd3121f728ff40706cdf2a1ef8ac8e18a6 (patch) | |
tree | 881d1f166d99b1ce669d8d0da62fd01de24c9dbf /src/cairo-pen.c | |
parent | 46dd21e795549481d9db8d90c399e683ef1205c7 (diff) | |
download | cairo-c7a35fbd3121f728ff40706cdf2a1ef8ac8e18a6.tar.gz |
use correctly transposed version of the matrix and fix up the comments above to use row vector notation.
Diffstat (limited to 'src/cairo-pen.c')
-rw-r--r-- | src/cairo-pen.c | 26 |
1 files changed, 14 insertions, 12 deletions
diff --git a/src/cairo-pen.c b/src/cairo-pen.c index 18b9ddb59..cad09fa6a 100644 --- a/src/cairo-pen.c +++ b/src/cairo-pen.c @@ -192,7 +192,7 @@ The letter t is used to represent the greek letter theta. 2. The question has been posed: What is the maximum expansion factor achieved by the linear transformation -X' = _R_ X +X' = X _R_ where _R_ is a real-valued 2x2 matrix with entries: @@ -246,7 +246,9 @@ circle on which X is constrained is to be parameterized by t: Thus - X'(t) = (a*cos(t) + b*sin(t), c*cos(t) + d*sin(t)) . + X'(t) = X(t) * _R_ = (cos(t), sin(t)) * [a b] + [c d] + = (a*cos(t) + c*sin(t), b*cos(t) + d*sin(t)). Define @@ -254,22 +256,22 @@ Define Thus - r^2(t) = (a*cos(t) + b*sin(t))^2 + (c*cos(t) + d*sin(t))^2 - = (a^2 + c^2)*cos^2(t) + (b^2 + d^2)*sin^2(t) - + 2*(a*b + c*d)*cos(t)*sin(t) + r^2(t) = (a*cos(t) + c*sin(t))^2 + (b*cos(t) + d*sin(t))^2 + = (a^2 + b^2)*cos^2(t) + (c^2 + d^2)*sin^2(t) + + 2*(a*c + b*d)*cos(t)*sin(t) Now apply the double angle formulae (A) to (C) from above: r^2(t) = (a^2 + b^2 + c^2 + d^2)/2 - + (a^2 - b^2 + c^2 - d^2)*cos(2*t)/2 - + (a*b + c*d)*sin(2*t) + + (a^2 + b^2 - c^2 - d^2)*cos(2*t)/2 + + (a*c + b*d)*sin(2*t) = f + g*cos(u) + h*sin(u) Where f = (a^2 + b^2 + c^2 + d^2)/2 - g = (a^2 - b^2 + c^2 - d^2)/2 - h = (a*b + c*d) + g = (a^2 + b^2 - c^2 - d^2)/2 + h = (a*c + b*d) u = 2*t It is clear that MAX[ |X'| ] = sqrt(MAX[ r^2 ]). Here we determine MAX[ r^2 ] @@ -377,12 +379,12 @@ _cairo_pen_vertices_needed (double tolerance, double a = matrix->xx, b = matrix->yx; double c = matrix->xy, d = matrix->yy; - double i = a*a + c*c; - double j = b*b + d*d; + double i = a*a + b*b; + double j = c*c + d*d; double f = 0.5 * (i + j); double g = 0.5 * (i - j); - double h = a*b + c*d; + double h = a*c + b*d; /* * compute major and minor axes lengths for |